Volumes of two-bridge cone manifolds in spaces of constant curvature
The author considers hyperbolic, Euclidean and spherical cone-manifolds $\Sigma(\alpha)$ whose underlying topological space is the 3-sphere and whose singular set, with a cone-angle $\alpha$ around it ($0 < \alpha \le 2\pi$), is a knot $\Sigma$ in $S^3$. If $\Sigma$ is a hyperbolic knot (i.e., has hyperbolic complement of finite volume) then, as a consequence of Thurston's hyperbolic Dehn surgery theorem, for all sufficiently small cone-angles of the form $\alpha = 2\pi/n$, the cone-manfold $\Sigma(\alpha)$ is also hyperbolic (in fact, a hyperbolic orbifold). After shortly discussing the situation for the trefoil and other torus knots, the first hyperbolic knot considered in the paper is the figure-8 knot $4_1$. Concerning cone-structures, the cone-manifold $4_1(\alpha)$ is hyperbolic for $0 < \alpha < 2\pi/3$, Euclidean for $\alpha = 2\pi/3$ (the 3-fold cyclic branched covering of the figure-8 knot is the Euclidean Hantzsche-Wendt manifold, cf. a paper by the reviewer in [Monatsh. Math. 110, No. 3--4, 321--327 (1990; Zbl 0717.57005)]), and spherical for $2\pi/3 < \alpha < 4\pi/3$. The first main results of the present paper are calculations of the volumes of these geometric cone-manifolds, and similar calculations are then done also for the other ten hyperbolic knots with at most seven crossings. All these knots are 2-bridge knots, so their 2-fold branched coverings are lens spaces and their cone-manifolds for $\alpha = \pi$ are spherical orbifolds (incidentally, with the only exception of the figure-8 knot, for all other hyperbolic knots the associated cone-manifolds (orbifolds) for $\alpha = 2\pi/n$ are hyperbolic, for $n \ge 3$, by the orbifold geometrization and Dunbar's list of the geometric, non-hyperbolic 3-orbifolds [\textit{W. D. Dunbar}, Rev. Mat. Univ. Complutense Madr. 1, No. 1--3, 67--99 (1988; Zbl 0655.57008)]). ``For 2-bridge knots with not more than 7 crossings, we present trigonometrical identities involving the lengths of singular geodesics and cone-angles of such cone-manifolds. Then these identities are used to produce exact integral formulae of the volume of the corresponding cone-manifold modeled in the hyperbolic, spherical and Euclidean geometries.'' Here the main tools for the volume calculations are the Schläfli formula and the $A$-polynomial for these manifolds. The paper serves also as a good survey on geometric cone-manifolds and orbifolds associated to knots, their volume-computations as well as on the relevant literature. (Question: Considering all hyperbolic 2-bridge knots, is the upper bound for hyperbolicity over all cone-angles equal to $\pi$?)
geometric cone-manifolds associated to 2-bridge knots
volumes of hyperbolic, Euclidean and spherical cone-manifolds
scientific article; zbMATH DE number 7380380
scientific article